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 A129907 Greatest prime factor of the number of subsets S of the power set P{1,2,...,n} such that: {1}, {2},..., {n} are all elements of S; if X and Y are elements of S and X and Y have a nonempty intersection, then the union of X and Y is an element of S. 0

%I

%S 2,3,7,683,143328791

%N Greatest prime factor of the number of subsets S of the power set P{1,2,...,n} such that: {1}, {2},..., {n} are all elements of S; if X and Y are elements of S and X and Y have a nonempty intersection, then the union of X and Y is an element of S.

%C The references are about the notion of connectivity spaces (in French, "espaces connectifs"): the sets S are the finite connectivity structures. For example, the set {1, 2, 3} in the above example is the Borromean structure. The computation of a(6) is entirely based on the work of Wim van Dam (cf. A072446).

%D R. Borger, Connectivity spaces and component categories, Categorical topology, International Conference on Categorical Topology, Berlin, Heldermann, 1984.

%D G. Matheron and J. Serra, Strong filters and connectivity, in Image Analysis and Mathematical Morphology 2, London, Academic Press, 1988, pp. 141-157.

%H Wim van Dam, <a href="http://www.cs.ucsb.edu/~vandam/research/spssequences.html">SubPower Set Sequences</a>.

%H S. Dugowson, <a href="http://msh.revues.org/document3908.html">Les frontieres dialectiques</a>, Mathematiques et sciences humaines, no. 177, Spring 2007.

%H S. Dugowson, <a href="http://arXiv.org/abs/0707.2542">Representation of finite connective spaces</a>, arXiv:0707.2542 [math.GN], 2007.

%e a(3)=3 because of the 12=3*2^2 subsets: {{1}, {2}, {3}}; {{1}, {2}, {3}, {1, 2}}; {{1}, {2}, {3}, {1, 3}}; {{1}, {2}, {3}, {2, 3}}; {{1}, {2}, {3}, {1, 2, 3}}; {{1}, {2}, {3}, {1, 2}, {1, 2, 3}}; {{1}, {2}, {3}, {1, 3}, {1, 2, 3}}; {{1}, {2}, {3}, {2, 3}, {1, 2, 3}}; {{1}, {2}, {3}, {1, 2}, {1, 3}, {1, 2, 3}}; {{1}, {2}, {3}, {1, 2}, {2, 3}, {1, 2, 3}}; {{1}, {2}, {3}, {1, 3}, {2, 3}, {1, 2, 3}}; {{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}.

%Y Cf. A072446.

%K more,nonn,uned

%O 2,1

%A S. Dugowson (dugowson(AT)ext.jussieu.fr), Jun 08 2007

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